+750 Find one ordered pair $(x,y)$ of real numbers such that $x + y = 10$ and $x^3 + y^3 = 162 + x^2 + y^2.$
+135 You might want to double check the question, because there is no real pair of numbers that holds those properties. There are, however, complex numbers that do:
y=10−x
x3+(10−x)3=162+x2+(10−x)2
x3+(10−x)2(10−x)=162+x2+(100−20x+x2)
x3+(100−20x+x2)(10−x)=2x2−20x+262
x3+(1000−200x+10x2−100x+20x2−x3)=2x2−20x+262
30x2−300x+1000=2x2−20x+262
15x2−150x+500=x2−10x+131
14x2−140x+369=0
x2−10x+36914=0
x2−10x+35014=−1914
x2−10x+25=−1914
(x−5)2=−1914
x−5=√1914i
x=5+√1914i
y=10−(5+√1914i)
y=5−√1914i
Therefore, one ordered pair of complex numbers that works is 5 + sqrt(19/14)i and 5 - sqrt(19/14)i
Please let me know if I had made any errors!
